optimal tracking control for unmanned aerial vehicles

165 Pages·2013·2.62 MB·English

OPTIMAL TRACKING CONTROL FOR
UNMANNED AERIAL VEHICLES
BY
´
PETER BAUER
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF
DOCTOR OF PHILOSOPHY
SUPERVISOR
Prof. J´ozsef Bokor
SYSTEMS AND CONTROL LABORATORY
Computer and Automation Research Institute
and
DEPARTMENT OF CONTROL FOR TRANSPORTATION- AND VEHICLE
SYSTEMS
Budapest University of Technology and Economics
BUDAPEST, HUNGARY
May 2013.
NYILATKOZAT
Alul´ırott Bauer P´eter kijelentem, hogy ezt a doktori ´ertekez´est magam k´esz´ıtettem ´es ab-
ban csak a megadott forr´asokat haszna´ltam fel. Minden olyan r´eszt, amelyet szo´ szerint,
vagy azonos tartalomban, de ´atfogalmazva m´as forr´asbo´l ´atvettem, egy´ertelmu˝en, a forr´as
megad´as´aval megjelo¨ltem.
Budapest, 2013. m´ajus 2.
————————————————
Bauer P´eter
Az ´ertekez´esro˝l k´eszu¨lt b´ıra´latok ´es a jegyz˝oko¨nyv a k´eso˝bbiekben a Budapesti Mu˝szaki
´es Gazdas´agtudoma´nyi Egyetem Ko¨zleked´esm´erno¨ki ´es J´armu˝m´erno¨ki Kara´nak D´ek´ani
Hivatala´ban el´erheto˝ek.
Acknowledgement
I would like to express my deepest gratitude to my advisor, Professor J´ozsef Bokor for
his patronage, technical guidance, many suggestions and constant encouragement during
this research.
I also wish to thank Bala´zs Kulcsa´r for his help at the beginning, and Ba´lint Vanek
for his help in the last years. I would like to thank the discussions, useful ideas, revision
of my articles and encouragement.
I had the pleasure of meeting Professor Gary Balas and the staffs of his control group
during my visits at Department of Aerospace Engineering and Mechanics at University
of Minnesota or during their visits in Hungary. I’m especially thankful to Pete Seiler,
Paw Yew Chai, Rohit Pandita, Andrei Dorobantu, Will Johnson and Arion Mangio for
working together.
Regarding the Department of Control for Transportation- and Vehicle Systems I’am
thankful to my colleagues Ka´roly Kurutz, G´eza Tarnai, Istv´an Hrivna´k, G´eza Szab´o,
Zolta´n Kom´ocsin, Ka´roly Gyenes, Bala´zs S´aghi, Edit Baranyi and Istv´an Varga for their
support and kindest help during my work at the department. I’am also thankful to the
others, Zsuzsa Preitl, Imre Ka´rolyi, Tam´as B´ecsi, Szil´ard Aradi, Tam´as Luspay, Tam´as
Tettamanti, Bala´zs N´emeth, Alfr´ed Csik´os and Andra´s Miha´ly for working and teaching
together. IwouldliketoexpressmyspecialthankstoErzs´ebetPaczonaforhercontinuous
administrative support and for the encouragement during the hard periods.
Regarding the Systems and Control Laboratory at Computer and Automation Re-
search Institute I’m very thankful to Alexandros Soumelidis for working together on
aerospace applications and making many discussions about electronics and microcon-
troller programming. I’m also thankful to the other colleagues P´eter Ga´spa´r, Andra´s
Edelmayer and Tam´as Bartha. I’am thankful to working together with Gergely Regula,
´
Ad´am Bakos, Gyula Pint´er, Istv´an Go˝zse, Ma´rk Luka´tsi, Andra´s Erd˝o and Istv´an R´eti.
I wish to express my thanks to my first roommates Tam´as P´eni, Ga´bor Ro¨d¨onyi and
Andra´s Bo´dis-Szomoru´ for their encouragement in research. I also would like to thank
the friendly atmosphere and encouragement to my current roommates Zolta´n Fazekas,
Zsolt Biro, Krisztia´n Szab´o and Ern˝o Simonyi. I wish to express my special thanks to
´
Katalin Hargitai, Marika Farkas and Eva Thiry for their continuous administrative sup-
port and for the encouragement during the hard periods.
At last but not at least, thanks to my parents for their constant support during the
first part of my work and for my wife Reka and her family for the same support during
the last part of my work.
Soli Deo Gloria
Notations and symbols
Notations
a: exponent of exponentially bounded reference signal / acceleration value
a: acceleration vector
A: maximum bound of exponentially bounded reference signal
A, B, C, C , G, V, W: discrete time system matrices / transformed discrete time
r
system matrices
Aa, Ba: matrices of system augmented by the observer
Ac, Bc, Gc, Cc: continuous time augmented system matrices
a a a r
Ac, Bc, Cc, Cc, Gc, Vc, Wc: continuous time system matrices
r
A , B , C , D : state space matrices of actuator delay dynamics
d d d d
A , B , C , D : state space matrices of washout filter
F F F F
A , B , C , D : uncertain discrete time system matrices
i i i i
A , C : time-varying state space matrices
k k
Ad: disturbance state dynamics matrix
y
A , A ,A, L : Heun scheme gain matrices
1 2 1
AW: anti windup gain
b: angular rate bias vector / wingspan
B : disturbance input matrix
d
B: augmented input matrix
c , c : roll / yaw moment coefficients
L N
C: projection matrix constructed from C
r
ˇ
C : transformed C matrix
r r
i
d: deterministic disturbance vector / dimension of disturbance vector / exponent of
disturbance exponential bound
D: maximum bound of exponentially bounded disturbance
˜
d: augmented disturbance vector
D , d : exponentially bounded state estimation error maximum bound and exponent
E E
d , d : roll and yaw torque disturbances
L N
d : exponent of exponentially bounded augmented disturbance
t
e,/e : output tracking error / roll angle tracking error
φ
e : unit gravity vector
G
e : integral of roll angle tracking error
I
E , E , E , E , E : error and signal norm values
φ r p δa δr
F: steady state input to output transition matrix / augmented matrix for feedforward
control design
G : aircraft dynamical model
ac
G : actuator dynamics (transfer function)
act
G : disturbance transfer matrix
d
G : actuator delay dynamical model
delay
Gd: disturbance to estimated disturbance transfer matrix
d
Ge: error to estimated disturbance transfer matrix
d
Gr: reference to estimated disturbance transfer matrix
d
G : estimation error transfer matrix
e
G : estimator EKF dynamics
est
G : smoothing filter for estimated disturbances
efilt
G : transfer function matrix including washout filter
filt
G : reference signal transfer matrix
r
Gd: disturbance to state transfer matrix
x
Ge: error to state transfer matrix
x
Gr: reference to state transfer matrix
x
ii
H: transformation matrix from reference to tracking state
H , H : prediction and control horizons in MPC control
p u
I , ω : engine and propeller inertia and angular rate
p p
J: cost functional / Jordan matrix
k: time index
K: gain of disturbance estimator
K : gain from disturbance
d
K : maximum bound of exponentially bounded augmented disturbance
D
K : gain from steady state disturbance
d∞
K : tracking error integral gain
I
K : reference feedforward gain
r
K , K : modified reference gains in MPC control
r1 r2
K : gain from steady state reference
r∞
K , K , K : gain matrices related to S /S matrices
S1 S2 S1 1 2
K : state feedback gain
x
K , K , K : optimal input gains (minimax control)
x Q S
K : pre-stabilizing state feedback gain
x1
K : LQ optimal state feedback gain
x2
K : maximum bound of exponentially bounded reference input / MPC gain from pre-
u
vious input
L: matrix of closed-loop estimation error dynamics
L , L , L , L , L , L : coefficients of aircraft linearized roll dynamics
p r δa δr dL dN
L : observer gain matrix
o
L , L , L : worst case disturbance gains (minimax control)
x Q S
m: dimension of input vector
M: constructor matrix for the linearly dependent columns of C / gain matrix in MPC
r
input design / gain of disturbance estimator
M , M , M , M : auxiliary matrices
2 3 4 R
iii
M : auxiliary input matrix
B
M+, M−: matrices of EKF state dynamics
n: dimension of system state space
N: control horizon / system noise covariance matrix
N , N , N , N , N , N : coefficients of aircraft linearized yaw dynamics
p r δa δr dL dN
N , N : gains of feedforward control
x u
p: dimension of measured output / roll rate
p : poles of open-loop system
0
p , p : varied parameter vectors
LQ MM
P: solution of ARE / DARE
PI: probability of instability
P : probability of saturation
Sat
P , I , D : gains of PID control
φ φ φ
q: dynamic pressure
Q, Q , Q : weighting matrices of cost function
1 2
r: reference vector / dimension of reference vector / yaw rate
r : (2 )r 1 dimensional augmented reference vector
2
· ×
R: input weighting matrix / augmented input weighting matrix
Ra: angular rate noise covariance matrix
RnH, Rna, R : magnetic, acceleration and GPS measurement noise covariance ma-
GPS
trices
RnH, Rna: basic noise covariance matrices
R : sum of reference related inputs
k
R , R : input / disturbance weighting matrices in minimax design
u d
s : actual value of forcing function
R
S: wing surface
S, S , S : gain matrices in costate variable and forcing function
1 2
T, T : similarity transformation matrices
S
iv
T , ζ : actuator time constant and damping
act act
TBE: earth to body transformation (rotation) matrix
T : system time delay
d
T : estimator EKF time lag
e
T , T : settling times
s1 s2
u: input vector / exponent of exponentially bounded reference inputs
ur: reference related part of input vector
v,w: stochastic disturbance vectors / dimensions of stochastic disturbance vectors
v: vector of measurements
v̺: angular rate noise vector
V: matrix from the linearly independent columns of C / velocity vector
r
V : wind strength
w
V , θ : trim velocity and pitch angle
0 0
W: measurement noise covariance matrix
x: state vector
xa: augmented system state vector
xd: state vector of actuator delay state space model / state vector of disturbance esti-
mator
xe: state estimation error vector
xF: state vector of washout filter state space model
x˜: tracking state vector
xˆ: estimated state vector
xˇ: transformed state vector
x: predicted state vector
xI, xII: parts of partitioned state vector
X: bidiagonal matrix of Jordan blocks
X, Y, Z: axes of coordinate systems
y: measured output vector
v
yr: tracking output vector
Y: diagonal block of Q weighting matrix
zˆ: predicted output in MPC control
Z matrix from forcing function system of equations
Z, T, ∆U vectors in MPC control prediction model
Z , Z : full block matrices in DARE solvability derivation
1 2
Greek notations
β: angle of sideslip
δ : aileron deflection
a
δ : rudder deflection
r
∆: difference from steady state
∆t: discrete time step
ε: MPC control tracking error vector
φ: roll (bank) angle
Φ: stabilized state space matrix
Φ : closed-loop state matrix
1
Φ : uncertain closed-loop state matrix
1i
Υ, Θ, Γ, Ω, Q, R: MPC prediction model matrices
λ: costate variable / eigenvalue
Λ: matrix in DARE solvability check
ψ: azimuth angle
ψ : wind direction
w
ρ,ρ: angular rate parameter vectors
θ: pitch angle
̺: quaternion vector
ω: angular velocity vector
vi
Acronyms
ARE: algebraic Riccati equation
DARE: discrete algebraic Riccati equation
EKF: extended Kalman filter
HIL: hardware-in-the-loop simulation
IAS: indicated airspeed
LQ: linear quadratic
MM: minimax
MPC: model predictive control
PID: proportional, integral, derivative control
UAV: unmanned aerial vehicle
UDE: unknown disturbance estimator
UDEB: unknown disturbance estimator through input matrix
Symbols
()+: Moore-Penrose pseudoinverse of a matrix / superscript of matrix in EKF dynamics
()E, ()B, ()N: notation in earth, body or normal coordinate systems
¯
(): mean value
()T: transpose of a matrix
vii